Hypothesis Test 1. A consumer group, concerned about the mean fat content of a certain grade of steak burger submits to an independent laboratory a random sample of 12 steak burgers for analysis. The The percentage percentage of fat fat in in each of the steak burgers is as follows: 21 18 19 1 18 2! 22 19 2! 1! 18 1" The manufacturer claims that the mean fat content of this grade of stea steak k burg burger er is less less than than 2#$. 2#$. Assu Assumi ming ng perc percent entag age e fat fat cont conten entt to be norm normal ally ly distributed with a standard de%iation of &, carry out an appropriate hypothesis test in order to ad%ise the consumer group as to the %alidity of the manufacturer's claim
2. (uring a particular week, 1& babies were born in a maternity unit. )art of the standard procedure is to measure the length of the baby. *i%en below is a list of the lengths, in centimeters, of the babies born in this particular week. !9 "# !" "1 !+ !9 !8 "! "& "" !" "# !8. Assuming Assuming that this sample sample came came from from an underlyi underlying ng normal normal populatio population, n, test, at the "$ signicance le%el, the hypothesis that the population mean length is "# cm.
3. A random sample of 12 steel ingots was taken from a production line. The masses, in kilograms, of these ingots are gi%en below. below. 2!.8 &#.8 28.1 2!.8 2+.! 22.1 2!.+ 2+.& 2+." 2+." 2+.8 2+.8 2&.9 2&.9 2&.2 2&.2 Assum Assuming ing that that this this samp sample le came came from from an an under underlyi lying ng normal normal population, in%estigate the claim that its mean e-ceeds 2".# kg. / 1$
4. A car manufacturer introduces a new method of assembling a particular component. The old method had a mean assembly time of !2 minutes. The manufacturer would like the assembly time to be as short as possible, and so he e-pects the new method to ha%e a smaller mean. A random sample of assembly times 0minutes taken after the new method had become become established established was 2+ &9 28 !1 !+ !2 &" &2 &8 tating tating any necessary necessary distributional assumptions, in%estigate the manufacturer's e-pectation. / 2 $
5. A random sample of 1" workers from a %acuum 3ask assembly line was selected from a large number of such workers. 4%or topwatch, a work5study engineer, asked each of these workers to assemble a one5litre %acuum 3ask at their normal working speed. The times taken, taken, in in seconds, seconds, to compl complete ete these tasks are gi%en gi%en below below:: 1#9.2 1#9.2 1!.2 1!.2 12+.9 12+.9 92.# 1#8." 91.1 1#9.8 11!.9 11".& 99.# 112.8 1&#.+ 1!1.+ 122. 119.9 Assuming that this sample came from an underlying normal population, in%estigate the claim that the population mean assembly time is less than 2 minutes. / "$
6. A pharmacist claims that more than #$ of all customers simply collect a prescription. 6ne of her assistants notes that, in a random sample of 12 customers, 1# simply collected a pres prescr crip ipti tion on.. (oes (oes this this pro% pro%id ide e su7c su7cie ient nt e%id e%iden ence ce,, at the the "$ le%e le%el, l, to suppo support rt the the pharmacist's claim
7. 4n a sur%ey carried out in un%ille, 1! children out of a random sample of &# said that they bought the opper comic regularly. Test, at the 1#$ le%el of signicance, the hypothesis that the true proportion of all children who buy this comic regularly is #.&".
8. A random sample of & coee drinkers were each asked to taste5test a new brand of coee. The responses are listed below with ; representing 'like', 4 representing 'indierent', and ( representing 'dislike'. ;(;;(;;;;;(( ;;;;;(;;;;(( ;;;;(;;;;;;( (o these data support the claim that more than half of all coee drinkers like this new brand of coee< / "$
9. )ackets of ground lter coee ha%e a nominal weight of 2## g. The distribution of weights may be assumed to be normal. A random sample of &2 packets had the following weights. 218 2#+ 21! 189 211 2# 2#& 21+ 18& 18 19+ 219 21& 2#+ 21! 2#& 2#! 19" 19+ 21& 212 188 221 21+ 18! 18 21 198 211 21 2## 2#8 4n%estigate the assumption that the mean weight of all packets is 2## g. Test the hypothesis that 1"$ of packets weigh less than 19# g. / &$
10.
=mployees of a rm carrying out motorway maintenance are issued with brightly colored waterproof >ackets. These come in dierent si?es numbered 1 to ". The last !# >ackets issued were of the following si?es. 2 & & 1 & & 2 ! & 2 " ! 1 2 & & 2 ! " & 2 ! ! 1 " & & 2 & & 1 & ! & & 2 " 1 ! ! Assuming that the !# employees may be regarded as a random sample of all employees, test the hypothesis, at the "$ signicance le%el, that !#$ of all employees re@uire si?e &. Test the claim that si?e & is the median si?e. 11. The Acme ompany has de%eloped a new battery. The engineer in charge claims that the new battery will operate continuously for at least + minutes longer than the old battery. To test the claim, the company selects a simple random sample of 1## new batteries and 1## old batteries. The old batteries run continuously for 19# minutes with a standard de%iation of 2# minutesB the new batteries, 2## minutes with a standard de%iation of !# minutes. Test the engineer's claim that the new batteries run at least + minutes longer than the old. Cse a #.#" le%el of signicance. 0Assume that there are no outliers in either sample.
12.
orty5four si-th graders were randomly selected from a sch ool district. Then, they were di%ided into 22 matched pairs, each pair ha%ing e@ual 4D's. 6ne member of each pair was randomly selected to recei%e special training. Then, all of the students were gi%en an 4D test. Test results are summari?ed below.
)air
Training Eo training
1
9"
9#
2
89
8"
&
+
+&
!
92
9#
"
91
9#
"&
"&
+
+
8
8
88
9#
9
+"
+8
1#
8"
89
11
9#
9"
Trainin
Eo
g
training
12
8"
8&
1&
8+
8&
1!
8"
8&
1"
8"
82
1
8
"
1+
81
+9
18
8!
8&
19
+1
#
2#
!
!+
21
+"
++
)air
22 8# 8& (o these results pro%ide e%idence that the special training helped or hurt student performance< Cse an #.#" le%el of signicance. Assume that the mean dierences are appro-imately normally distributed.
13.
Fithin a school district, students were randomly assigned to one of two Gath teachers 5 Grs. mith and Grs. Hones. After the assignment, Grs. mith had &# students, and Grs. Hones had 2" students. At the end of the year, each class took the same standardi?ed test. Grs. mith's students had an a%erage test score of +8, with a standard de%iation of 1#B and Grs. Hones' students had an a%erage test score of 8", with a standard de%iation of 1".
Test the hypothesis that Grs. mith and Grs. Hones are e@ually eecti%e teachers. Cse a #.1# le%el of signicance. 0Assume that student performance is appro-imately normal.
14.
The =6 of a large electric utility claims that 8# percent of his 1,###,### customers are %ery satised with the ser%ice they recei%e. To test this claim, the local newspaper sur%eyed 1## customers, using simple random sampling. Among the sampled customers, +& percent say they are %ery satisied. ased on these ndings, can we re>ect the =6's hypothesis that 8#$ of the customers are %ery satised< Cse a #.#" le%el of signicance.
15.
The =6 of a large electric utility claims that 8# percent of his 1,###,### customers are %ery satised with the ser%ice they recei%e. To test this claim, the local newspaper sur%eyed 1## customers, using simple random sampling. Among the sampled customers, +& percent say they are %ery satised. ased on these ndings, can we re>ect the =6's hypothesis that 8#$ of the customers are %ery satised< Cse a #.#" le%el of signicance.
16.
uppose the Acme (rug ompany de%elops a new drug, designed to pre%ent colds. The company states that the drug is e@ually eecti%e for men and women. To test this claim, they choose a simple random sample of 1## women and 2## men from a population
of 1##,### %olunteers. At the end of the study, &8$ of the women caught a coldB and "1$ of the men caught a cold. ased on these ndings, can we re>ect the company's claim that the drug is e@ually eecti%e for men and women< Cse a #.#! le%el of signicance.
17.
uppose the pre%ious e-ample is stated a little bit dierently. uppose the Acme (rug ompany de%elops a new drug, designed to pre%ent colds. The company states that the drug is more eecti%e for women than for men. To test this claim, they choose a simple random sample of 1## women and 2## men from a population of 1##,### %olunteers. At the end of the study, &8$ of the women caught a coldB and "1$ of the men caught a cold. ased on these ndings, can we conclude that the drug is more eecti%e for women than for men< Cse a #.#1 le%el of signicance.
18.
Cn diseIador de productos estJ interesado en reducir el tiempo de secado de una pintura. e prueban dos fKrmulas de pinturaB la fKrmula 1 tiene el contenido @uLmico estJndar y la fKrmula 2 tiene un nue%o ingrediente secante @ue tiende a reducir el tiempo de secado. (e la e-periencia se sabe @ue la des%iaciKn estJndar del tiempo de secado es ocho minutos y esta %ariabilidad inherente no debe %erse afectada por adiciKn del nue%o ingrediente. e pintan &" placas con la fKrmula 1 y otras &" con la fKrmula 2. ;os dos tiempos promedio de secado muestrales son 11 minutos para la fKrmula 1 y 112 minutos para la fKrmula 2. MA @uN conclusiKn puede llegar el diseIador del producto sobre la ecacia del nue%o ingrediente, al ni%el de signicancia #,#1<
19.
inco muestras de una sustancia ferrosa se usan para determinar si hay una diferencia entre un anJlisis @uLmico de laboratorio y un anJlisis de 3uorescencia de rayos O del contenido de hierro. ada muestra se di%ide en dos submuestras y se aplican los dos tipos de anJlisis. A continuaciKn, se presentan los datos codicados @ue muestran los anJlisis de contenido de hierro: uponga @ue las poblaciones son normales, pruebe con un ni%el de signicancia de #,#" si los dos mNtodos de anJlisis dan, en promedio, el mismo resultado O5ray hemical
2.# 2.# 2.& 2.1 2.! 2.2 1.9 2." 2.& 2.!
uponga @ue las poblaciones son normales, pruebe con un ni%el de signicancia de #,#" si los dos mNtodos de anJlisis dan, en promedio, el mismo resultado.
20.
Una muestra de "# familias de una comunidad muestra @ue 1# de ellas estJn %iendo
un programa especial de tele%isiKn sobre la economLa nacional. =n una segunda comunidad 1" familias de una muestra aleatoria de "# estJn %iendo el programa especial de tele%isiKn, a continuaciKn se prueba la hipKtesis de @ue la proporciKn general de tele%identes en las dos comunidades no diere, usando el ni%el de signicancia de 1$
21.
e ponen a prueba la enseIan?a de la =stadLstica empleando =-cel y Finstats. )ara determinar si los estudiantes dieren en tNrminos de estar a fa%or de la nue%a enseIan?a se toma una muestra aleatoria de 2# estudiantes por cada paralelo. (el paralelo A 18 estJn a fa%or, en tanto @ue del paralelo estJn a fa%or 1!. M=s posible concluir con un
ni%el de signicaciKn de #,#" @ue los estudiantes @ue estJn a fa%or de la nue%a enseIan?a de la =stadLstica es la misma en los dos paralelos<.
PRUEBA CHI CUDRAD 22.
upongamos @ue un in%estigador estJ interesado en e%aluar la asociaciKn entre @uienes poseen %ehLculo propio y el ni%el socioeconKmico del conductor. on este ob>eto se toma una muestra de conductores a @uienes se clasica en una tabla de asociaciKn, encontrando los siguientes resultados:
Posee !eh"#$ %o p&opio
'i!e% 'i!e% 'i!e% so#ioe#o()* so#ioe#o()* so#ioe#o()* i#o +,-o i#o *eio i#o ,%to
4 E6
8 1" 1& 1 T,+%, I. Tabla de asociaciKn, %alores obser%ados.
28 1!
M)ermiten estos datos armar @ue el uso del cinturKn de seguridad depende del ni%el socioeconKmico< Csar un ni%el de signicaciKn alfa/#,#".
23.
Tomemos como e>emplo la distribuciKn esperada para los indi%iduos de una poblaciKn @ue son clasicados segPn grupo sanguLneo. egPn estudios reali?ados en poblaciKn, se espera @ue dicha distribuciKn, en porcenta>es, sea la siguiente:
/&$po
&e#$e(#i, espe&,,
A A #
2,#$ &#,"$ 9,&$ "8,2$
=n una muestra de 1"# donadores de sangre se encontrK la siguiente distribuciKn:
/&$po
&e#$e(#i, o+se&!,,
A A #
! !8 1" 8&
e a>ustan los datos obser%ados a la distribuciKn teKrica< / 1$
24.
Huan GNnde?, director de Gercadeo de Aladino, tiene la responsabilidad de controlar el ni%el de e-istencias para cuatro tipos de automK%il %endidos por la rma. =n el pasado, ha ordenado nue%os automK%iles ba>o la premisa de @ue los cuatro tipos son igualmente populares y la demanda de cada tipo es la misma. in embargo, recientemente las e-istencias se han %uelto mJs difLciles de controlar. Huan considera @ue deberLa probar su
hipKtesis respecto a una demanda uniforme. ;a demanda es uniforme para los cuatro tipos de autos< Tipo de auto
Kia Fiesta Focus Clio
Ventas observadas 15 11 10 12
Ventas esperadas 12 12 12 12
25.
)aty Al%arado es la directora de in%estigaciKn de )laguicidas. =n su proyecto actual )aty debe determinar si e-iste alguna relaciKn entre la clasicaciKn de efecti%idad @ue los consumidores asignan a un nue%o insecticida y el sitio 0urbano o rural en el cual se utili?a. (e los 1## consumidores a @uienes se le aplicK la encuesta, +" %i%Lan en ?onas urbanas y 2" en ?onas rurales. ;a Tabla 1.2 resume las clasicaciones hechas por los consumidores. Clasificación Arriba del promedio
Promedio Debajo del promedio
Urbano fo = 20
Rural fo =11
fo = 40
fo = 8
fo = 15
fo = 6
;a clasicaciKn y la ubicaciKn son independientes<
26.
A Cni%ersity conducted a sur%ey of its recent graduates to collect demographic and health information for future planning purposes as well as to assess students' satisfaction with their undergraduate e-periences. The sur%ey re%ealed that a substantial proportion of students were not engaging in regular e-ercise, many felt their nutrition was poor and a substantial number were smoking. 4n response to a @uestion on regular e-ercise, #$ of all graduates reported getting no regular e-ercise, 2"$ reported e-ercising sporadically and 1"$ reported e-ercising regularly as undergraduates. The ne-t year the Cni%ersity launched a health promotion campaign on campus in an attempt to increase health beha%iors among undergraduates. The program included modules on e-ercise, nutrition and smoking cessation. To e%aluate the impact of the program, the Cni%ersity again sur%eyed graduates and asked the same @uestions. The sur%ey was completed by !+# graduates and the following data were collected on the e-ercise @uestion:
Eumber of tudents
Eo Qegular =-ercise
poradic =-ercise
Qegular =-ercise
2""
12"
9#
ased on the data, is there e%idence of a shift in the distribution of responses to the e-ercise @uestion following the implementation of the health promotion campaign on campus< Qun the test at a "$ le%el of signicance.
27.
The Eational enter for Realth tatistics 0ER pro%ided data on the distribution of weight 0in categories among Americans in 2##2. The distribution was based on specic %alues of body mass inde- 0G4 computed as weight in kilograms o%er height in meters s@uared. Cnderweight was dened as G4S 18.", Eormal weight as G4 between 18." and 2!.9, o%erweight as G4 between 2" and 29.9 and obese as G4 of &# or greater. Americans in 2##2 were distributed as follows: 2$ Cnderweight, &9$ Eormal Feight, &$ 6%erweight, and 2&$ 6bese. uppose we want to assess whether the distribution of G4 is dierent in the ramingham 6spring sample. Csing data from the n/&,&2 participants who attended the se%enth e-amination of the 6spring in the ramingham Reart tudy we created the G4 categories as dened and obser%ed the following:
Cnderweight Eormal Feight
U )articipants
6%erweight
6bese
G4S18."
G4 18."52!.9
G4 2".#5 29.9
G4 & #
2#
9&2
1&+!
1###
of
et up the hypotheses and determine le%el of signicance "$
28.
4n a prior e-ample we e%aluated data from a sur%ey of uni%ersity graduates which assessed, among other things, how fre@uently they e-ercised. The sur%ey was completed by !+# graduates. 4n the prior e-ample we used the V 2 goodness5of5t test to assess whether there was a shift in the distribution of responses to the e-ercise @uestion following the implementation of a health promotion campaign on campus. Fe specically considered one sample 0all students and compared the obser%ed distribution to the distribution of responses the prior year 0a historical control. uppose we now wish to assess whether there is a relationship between e-ercise on campus and students' li%ing arrangements. As part of the same sur%ey, graduates were asked where they li%ed their senior year. The response options were dormitory, on5campus apartment, o5campus apartment, and at home 0i.e., commuted to and from the uni%ersity. The data are shown below.
Eo porad Qegular ic =-ercise =-ercis e
Qegular =-ercise
(ormitory
&2
&#
28
6n5ampus
+!
!
!2
Apartment
65ampus Apartment
At Rome
Total
11#
2"
1"
&9
"
2""
12"
9#
et up hypotheses and determine le%el of signicance. R#: ;i%ing arrangement and e-ercise are independentB /#.#"
29.
A randomi?ed trial is designed to e%aluate the eecti%eness of a newly de%eloped pain relie%er designed to reduce pain in patients following >oint replacement surgery. The trial compares the new pain relie%er to the pain relie%er currently in use 0called the standard of care. A total of 1## patients undergoing >oint replacement surgery agreed to participate in the trial. )atients were randomly assigned to recei%e either the new pain relie%er or the standard pain relie%er following surgery and were blind to the treatment assignment. efore recei%ing the assigned treatment, patients were asked to rate their pain on a scale of #51# with higher scores indicati%e of more pain. =ach patient was then gi%en the assigned treatment and after &# minutes was again asked to rate their pain on the same scale. The primary outcome was a reduction in pain of & or more scale points 0dened by clinicians as a clinically meaningful reduction. The following data were obser%ed in the trial.
T&e,t*e(t /&o$p
(
'$*+e& ith Re$#tio( o 3 Poi(ts
'e P,i( Re%ie!e&
"#
2&
t,(,& P,i( Re%ie!e&
"#
11
'$*+e& ith Re$#tio( o 3 Poi(ts
Test whether there was a signicant dierence in the proportions of patients reporting a meaningful reduction 0i.e., a reduction of & or more scale points using a W statistic, as follows. et up hypotheses and determine le%el of signicance "$
30.
=l director de una escuela clasica a los padres en tres categorLas socio5econKmicas segPn su Jrea de residencia y en tres ni%eles de participaciKn en acti%idades escolares. )robar la hipKtesis de @ue no e-iste relaciKn entre el ni%el socio5econKmico y la participaciKn en acti%idades escolares, con un ni%el de signicaciKn del !$ )articipaciK n Eunca 6casional Qegularme nte
Ei%el de ingreso a>o Gedio Alto 28 !8 1 22 " 1! 1+
+!
&
31.
Cna agencia de publicidad intenta determinar la composiciKn demogrJca del mercado para un nue%o producto. eleccionaron al a?ar +" personas de cada uno de " grupos de edad y les presentaron el producto. ;os resultados de la encuesta son los siguientes: Actitud frente al producto ompra frecuente ompra rara %e? Eunca compra
*rupo de edad 18 5 29
&# 5 &9 !# 5 !9 "# 5 "9 # 5 9
12
18
1+
22
&2
18 !"
2" &2
29 29
2! 29
&# 1&
, (esarrolle una tabla de frecuencias obser%adas y esperadas para este problema + alcule el %alor V2 de la muestra. # =stable?ca las hipKtesis nula y alternati%a. i el ni%el de signicancia es #.#1, Mdebe recha?arse la hipKtesis nula<
32.
(espuNs de aIos de traba>ar en una estaciKn de pesado para camiones, He impson siente @ue el peso por camiKn 0en miles de libras sigue una distribuciKn normal. on el ob>eto de probar esta suposiciKn, He recolecta los siguientes datos un lunes y registra el peso de cada camiKn @ue llega a su bJscula. 85 57 60 81 89 63
89 50 63 76
52 86 95 56 61 75 50 66
65 90 78 95 81 50 62 97
77 60 66 60 61 98 79 67
64 57 92 82 53 63 69 54
61 55 77 93
i He usa la prueba de bondad de a>uste de >i5cuadrada para estos datos, M@uN concluye acerca de la distribuciKn del peso de los camiones< 0Cse un ni%el de signicancia de #.1# y asegPrese de establecer la hipKtesis de interNs. 0 Sugerencia: use cinco inter%alos igualmente probables.
33.
=l coordinador de computaciKn en la escuela de administraciKn cree @ue el tiempo @ue un estudiante de posgrado dedica a leer y escribir correos electrKnicos cada dLa de la semana tiene una distribuciKn normal. )ara e-aminar esta opiniKn, el coordinador recolecta datos un miNrcoles y registra el tiempo en minutos @ue cada estudiante gasta es sus correos electrKnicos. Cse la prueba de bondad de a>uste de >i5cuadrada con estos datos, M@uN concluye acerca de la distribuciKn del tiempo dedicado al correo electrKnico< 0Ctilice #.1# para el ni%el de signicancia y estable?ca con claridad sus hipKtesis. 8.2 12.! 1.2 19.! 12.& 1&.9 1!.& 12.!
+.! 9. 12.8 1#. 18. &.& 1".+ 12.8 2#.! 11.& 1#.9 18.! 18.& 19.2 1!.9 1.+ 11.& 18.1 2#.1
22.! 18.!
.2 12.!
8.+
9.+
1".9
1!.&
1.2
.+
18.!
18.8
2#.!
34.
;a siguiente tabla recoge informaciKn sobre una encuesta reali?ada a una muestra de hogares de " comunidades diferentes respecto a la audiencia del programa radial XRabla el sabelotodoY. )robar la hipKtesis nula de @ue no e-iste diferencia signicati%a en la audiencia entre los hogares de las " ?onas. / !$
35.
=scuchK el programa Eo escuchK el programa
A
Wonas
(
=
12
1+
8
21
2"
&+
&#
!1
2+
1
=l director de una escuela clasicK a los padres de familia en & categorLas socio5econKmicas y en & ni%eles de participaciKn en acti%idades de la escuela. probar la hipKtesis nula de @ue no e-iste relaciKn entre el ni%el socio5econKmico y la participaciKn en las acti%idades escolares. / &$ )articipaciKn en acti%idades Eunca 6casional recuenteme nte
Ei%el socio5 econKmico a>o Gedio Alto 28 !8 1 22 " 1! 1+
+!
&
RE/REI' CRREACI'
1.
;as notas de 12 alumnos de una clase en GatemJticas y Lsica son las siguientes: GatemJticas 2 & ! ! " + + 8 1# 1# Lsica 1 & 2 ! ! ! ! + 9 Rallar el coeciente de correlaciKn de la distribuciKn e interpretarlo.
1#
2. Cna compaILa de seguros considera @ue el nPmero de %ehLculos @ue circulan por una determinada autopista a mJs de 12# kmZh, puede ponerse en funciKn del nPmero de accidentes @ue ocurren en ella. (urante " dLas obtu%o los siguientes resultados:
A##ie(tes ':*e&o e !eh"#$%os a b c d e
3.
"
+
2
1
9
1" 18 1# 8 2# alcula el coeciente de correlaciKn lineal. i ayer se produ>eron accidentes, cuJntos %ehLculos podemos suponer @ue circulaban por la autopista a mJs de 12# km Z h< =s buena la predicciKn< i ayer circularon 1+ %ehLculos a mJs de 12# [mZhora, cuJntos accidentes se esperarLa tener =labore un inter%alo de conan?a para el literal b, utili?ando un ni%el de conan?a del 9&$
=l nPmero de obreros 0en millones ocupados en la agricultura, para los aIos @ue se indican, era: AIo
2##+ 2##8 2##9 2#1# 2#11 2#12 2#1&
2#1 !
6cupados 2,1 2,#! 1,9 1,+! 1,9 1,!9 1,2" 1,1 , )odrLa e-plicarse su e%oluciKn mediante una recta de regresiKn< + DuN limitaciones tendrLan las estimaciones hechas por esa recta< # uJl es el nPmero de traba>adores @ue se esperarLa tener para el aIo 2#1" y 2#1< 4nter%alo de conan?a 09#$ para la estimaciKn del aIo 2#1" e =s able la estimaciKn<
4.
Asocia las rectas de regresiKn y / \- ]1, y / 2- \ 12, y / #,"- ] " a las nubes de puntos siguientes:
Asigna los coecientes de correlaciKn lineal r / #,!, r / \#,8" y r / #,+, a las nubes del problema anterior.
5. ;a tabla siguiente muestra las notas obtenidas por 8 alumnos en un e-amen, las horas de estudio dedicadas a su preparaciKn y las horas @ue %ieron la tele%isiKn los dLas pre%ios al e-amen. Eota
"
+
&
"
8
!
9
Roras de estudio
+
1#
9
!
8
1#
"
1!
Roras de T^
+
2
11
9
&
9
"
, Qepresenta grJcamente los diagramas correspondientes a nota5estudio y nota5T^. + Me obser%a correlaciKn entre las %ariables estudiadas< M(e @uN tipo< M=n @uN caso estimas @ue es mJs fuerte< C Rallar el coeciente de correlaciKn de nota5estudio y nota5T^. MDuN puede deducirse con mJs precisiKn conociendo la nota @ue obtu%o una persona en el e-amen: el tiempo @ue dedicK al estudio o el @ue dedicK a %er la tele%isiKn< Q / #,9!&&82 y #,8!28&. Qespecti%amente. on los mismos datos, hallar las rectas de regresiKn correspondientes y estima para un alumno @ue sacK un 2 en el e-amen: las horas @ue estudiK y ;as horas @ue %io la T^.
6. (urante su primer aIo de %ida han pesado a Garta cada mes. =n la tabla siguiente se dan sus pesos:
Edad
1
2
Peso &,2 &,+
&
!
"
+
8
9
1#
11
12
!,2
",&
",+
,"
,8
+,2
+,9
+,+
8
8,"
, alcula la media y la des%iaciKn tLpica de los pesos.
+ (etermina la ecuaciKn de la recta de regresiKn de y sobre x , e-plicando detalladamente los cJlculos reali?ados y las fKrmulas @ue utili?as Q / a ,22"B 1,+181 b y / #,!8+#] &,#"9#9
# Rallar el coeciente de determinaciKn i Garta llega a tener 1! meses, @uN peso se esperarLa< e i se espera de Garta un peso de +,+ [g. DuN edad deberLa tener< i se espera de Garta un peso de 8,9 [g. DuN edad deberLa tener< ; 6btenga un inter%alo de conan?a 09&$ para la estimaciKn del literal e h 6btenga un inter%alo de conan?a 09!$ para la estimaciKn del literal h
7. =l dueIo de un restaurante de hamburguesas en la ciudad desea determinar la interrelaciKn entre la introducciKn de adere?os importados y las utilidades @ue recibe.
Ctilidades +# (emanda de cJtsup nacional 2 (emanda de cJtsup "# importada
!# 1 "
1## & +"
8# 2 &#
&# 1 !"
1## & &"
on esta informaciKn determinar lo siguiente:
, + #
;a ecuaciKn de regresiKn lineal mPltiple. ;a prueba de signicancia del modelo "$ 4nter%alos de conan?a del 9" $ para los parJmetros del modelo. 4nter%alos de conan?a del 9# $ para la utilidad esperada y la futura cuando la demanda de cJtsup nacional sea de ! y la de cJtsup importada de "#. e =l coeciente de determinaciKn mPltiple. i la demanda de cJtsup nacional es de ! y la importada es de "8B cuJles serLan las utilidades esperadas< ; i la demanda de cJtsup nacional es de & y la importada es de #B cuJles serLan las utilidades esperadas<
8. am Tadeus, dueIo y gerente general de un almacNn de e@uipos electrKnicos, estJ preocupado por el comportamiento de las %entas de un modelo de (^( @ue se %enden en la tienda. e da cuenta de @ue e-isten muchos factores @ue podrLan ayudar a e-plicarlo, pero cree @ue la publicidad y los %endedores son los principales determinantes. am reuniK los siguientes datos: ^entas en unidad es 1
E_ de %endedor *asto en es publicidad ! 1&&
&& 1 +# 82 1+ 2!
& 1# 1& 9
12" 11" 1!# 1&# 1!" 1!#
, alcule la ecuaciKn de mLnimos cuadrados para predecir las %entas a partir de la + # e
publicidad y los %endedores. i se gasta en publicidad es ` 1&8 y se emplean + %endedores, @uN %entas podrLa pronosticar< i se espera %ender !# e@uipos y se utili?a " %endedores, @uN gasto en publicidad se esperarLa< i se espera %ender !" e@uipo y se gasta ` 18#, cuJntos %endedores se esperarLa tener < alcular el error estJndar de la estimaciKn alcular el inter%alo de conan?a 09#$ para la estimaciKn del literal c
9. A small study is conducted in%ol%ing 1+ infants to in%estigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams.
, Fe wish to estimate the association between gestational age and infant birth weight. 4n this e-ample, birth weight is the dependent %ariable and gestational age is the independent %ariable.
+ uild a condence inter%al with 9" $, between eight and weight
# 4f the eight is &+.8, which should be the e-pected weight< 4f the weight is 299#, which should be the e-pected eight< e uild an condence inter%al 09#$ for the estimation of c. (etermine the correlation between these %ariables.
10.
The following table shows a researcher works with Al?heimers caregi%ers Table 1. Gade5up data for the predictors of scores for @uality of life.
a) Determie t!e correlatios of t!ese "ariables #it! eac! ot!er$
+ %et&s sa' t!at a perso !as a score o t!e measure of (ocial (upport of 20 ad t!e' !a"e 50)000 i fiacial assets$ *!at #ould 'ou +et if 'ou plu+ t!ese umbers ito t!e re+ressio e,uatio-
. = /$42
